Uniformly accelerated motion
Uniformly Accelerated Motion
Uniformly accelerated motion refers to the motion of an object that is moving under the influence of a constant acceleration. This means that the object's velocity changes at a constant rate over time. This type of motion is a key concept in classical mechanics and is often discussed in the context of objects moving under the influence of gravity or other constant forces.
Key Concepts
Before diving into the details of uniformly accelerated motion, let's define some key concepts:
- Displacement ($s$): The change in position of an object.
- Initial Velocity ($u$): The velocity of the object at the start of the time interval.
- Final Velocity ($v$): The velocity of the object at the end of the time interval.
- Acceleration ($a$): The rate at which the velocity of an object changes with time.
- Time ($t$): The duration over which the motion occurs.
Equations of Uniformly Accelerated Motion
The motion of an object under uniform acceleration can be described by the following equations, often referred to as the "equations of motion":
First Equation of Motion: [ v = u + at ] This equation relates the final velocity to the initial velocity, acceleration, and time.
Second Equation of Motion: [ s = ut + \frac{1}{2}at^2 ] This equation gives the displacement of the object in terms of initial velocity, acceleration, and time.
Third Equation of Motion: [ v^2 = u^2 + 2as ] This equation relates the final velocity to the initial velocity, acceleration, and displacement.
Table of Differences and Important Points
| Property | Uniformly Accelerated Motion | Non-uniformly Accelerated Motion |
|---|---|---|
| Acceleration | Constant | Variable |
| Velocity | Changes linearly with time | Changes non-linearly with time |
| Displacement | Quadratic function of time | Not necessarily quadratic |
| Equations of Motion | Applicable | Not applicable |
Examples
Example 1: Free Falling Object
A classic example of uniformly accelerated motion is a free-falling object under the influence of gravity near the Earth's surface, where the acceleration due to gravity ($g$) is approximately $9.81 \, \text{m/s}^2$.
Given: An object is dropped from rest (initial velocity $u = 0$) from a height of $45 \, \text{m}$.
Find: The time it takes to hit the ground and its velocity just before impact.
Solution:
Using the second equation of motion to find the time ($t$): [ s = ut + \frac{1}{2}gt^2 ] [ 45 = 0 \cdot t + \frac{1}{2} \cdot 9.81 \cdot t^2 ] [ t^2 = \frac{2 \cdot 45}{9.81} ] [ t = \sqrt{\frac{90}{9.81}} \approx 3.03 \, \text{s} ]
Using the first equation of motion to find the final velocity ($v$): [ v = u + gt ] [ v = 0 + 9.81 \cdot 3.03 ] [ v \approx 29.7 \, \text{m/s} ]
Example 2: Car Accelerating from Rest
A car accelerates uniformly from rest with an acceleration of $3 \, \text{m/s}^2$ for $5$ seconds.
Given: Initial velocity $u = 0$, acceleration $a = 3 \, \text{m/s}^2$, and time $t = 5 \, \text{s}$.
Find: The final velocity and the distance covered by the car.
Solution:
Using the first equation of motion to find the final velocity ($v$): [ v = u + at ] [ v = 0 + 3 \cdot 5 ] [ v = 15 \, \text{m/s} ]
Using the second equation of motion to find the displacement ($s$): [ s = ut + \frac{1}{2}at^2 ] [ s = 0 \cdot 5 + \frac{1}{2} \cdot 3 \cdot 5^2 ] [ s = \frac{1}{2} \cdot 3 \cdot 25 ] [ s = 37.5 \, \text{m} ]
Conclusion
Uniformly accelerated motion is a fundamental concept in kinematics, providing a framework for analyzing the motion of objects under constant acceleration. By understanding and applying the equations of motion, one can predict and analyze the behavior of such objects in various scenarios.